Moving objects whilst in animation
This isn't possible outside of the game engine, but within the BGE, you can interact with objects quite easily. You can set up your objects in Blender Render, but the physics are a little different in the BGE so it's better to just work in the latter. 

Start by setting up your objects and their physics properties. Then open the logic editor and set up either keyboard or mouse sensors for the moveable object (make sure it's selected) and attach the sensors to motion actuators. Set up the actuator's appropriate fields to tell it which way to move the object and how fast. That's about all there is to it. You can now start it as a game by putting the cursor in the 3D viewer and pressing P, then use your keys to move your object around.